# ST-PINN: A Self-Training Physics-Informed Neural Network for Partial Differential Equations

arXiv: https://arxiv.org/abs/2306.09389 · HTML: https://mlsanity.com/paper/2306.09389/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 33 references that are in the map.
- First posted on arXiv 15 Jun 2023.

## Builds on (most cited first, 15 of 15)

- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [Physics-informed neural networks (PINNs) for fluid mechanics: A review](https://mlsanity.com/paper/2105.09506/) (2021)
- [Semi-Supervised Semantic Segmentation with Cross Pseudo Supervision](https://mlsanity.com/paper/2106.01226/) (2021)
- [PDEBENCH: An Extensive Benchmark for Scientific Machine Learning](https://mlsanity.com/paper/2210.07182/) (2022)
- [Rethinking Pre-training and Self-training](https://mlsanity.com/paper/2006.06882/) (2020)
- [B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2003.06097/) (2020)
- [A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks](https://mlsanity.com/paper/2207.10289/) (2022)
- [Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems](https://mlsanity.com/paper/2111.02801/) (2021)
- [Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating a class of inverse problems for PDEs](https://mlsanity.com/paper/2007.01138/) (2020)
- [Generic bounds on the approximation error for physics-informed (and) operator learning](https://mlsanity.com/paper/2205.11393/) (2022)
- [On the convergence of physics informed neural networks for linear second-order elliptic and parabolic type PDEs](https://mlsanity.com/paper/2004.01806/) (2020)
- [Teaching the Incompressible Navier-Stokes Equations to Fast Neural Surrogate Models in 3D](https://mlsanity.com/paper/2012.11893/) (2020)
- [Physics-informed Spline Learning for Nonlinear Dynamics Discovery](https://mlsanity.com/paper/2105.02368/) (2021)
- [Physics-aware Spatiotemporal Modules with Auxiliary Tasks for Meta-Learning](https://mlsanity.com/paper/2006.08831/) (2020)

## Built on by (most cited first, 6 of 6)

- [Auxiliary-Tasks Learning for Physics-Informed Neural Network-Based Partial Differential Equations Solving](https://mlsanity.com/paper/2307.06167/) (2023)
- [Gradient Enhanced Self-Training Physics-Informed Neural Network (gST-PINN) for Solving Nonlinear Partial Differential Equations](https://mlsanity.com/paper/2510.10483/) (2025)
- [PHANTOM: Physics-Aware Adversarial Attacks against Federated Learning-Coordinated EV Charging Management System](https://mlsanity.com/paper/2512.22381/) (2025)
- [Enhancing Convergence Speed with Feature-Enforcing Physics-Informed Neural Networks: Utilizing Boundary Conditions as Prior Knowledge for Faster Convergence](https://mlsanity.com/paper/2308.08873/) (2023)
- [Can Physics Informed Neural Operators Self Improve?](https://mlsanity.com/paper/2311.13885/) (2023)
- [Physics and Data Driven Transformer-Mamba Framework for Flow Field](https://mlsanity.com/paper/2609.29087/) (2026)

## Abstract (arXiv)

Partial differential equations (PDEs) are an essential computational kernel in physics and engineering. With the advance of deep learning, physics-informed neural networks (PINNs), as a mesh-free method, have shown great potential for fast PDE solving in various applications. To address the issue of low accuracy and convergence problems of existing PINNs, we propose a self-training physics-informed neural network, ST-PINN. Specifically, ST-PINN introduces a pseudo label based self-learning algorithm during training. It employs governing equation as the pseudo-labeled evaluation index and selects the highest confidence examples from the sample points to attach the pseudo labels. To our best knowledge, we are the first to incorporate a self-training mechanism into physics-informed learning. We conduct experiments on five PDE problems in different fields and scenarios. The results demonstrate that the proposed method allows the network to learn more physical information and benefit convergence. The ST-PINN outperforms existing physics-informed neural network methods and improves the accuracy by a factor of 1.33x-2.54x. The code of ST-PINN is available at GitHub: https://github.com/junjun-yan/ST-PINN.